N90-293 1

THE EFFECTS OF GEAR REDUCTION ON ROBOT DYNAMICS

J. Chen

Department of Mechanical Engineering

University of Maryland

College Park, MD 20742

Abstract

The effect of the joint drive system with gear reduction for a generic

two-link system is studied. It is done by comparing the kinetic energy of

such a system with that of a direct drive two-link system. The onlydifference are two terms involving the inertia of the motor rotor and gear

ratio. Modifications of the equations of motion from a direct drive system

are then developed and generalized to various cases encountered in robot

manipulators.

Introduction

Formulating the equations of motion for a robot is an important part of

robot analysis that will provide necessary information for the design ofcontrol laws and mechanical components. Before the process of formulation can

begin, idealization of the robot system into a model amenable to analysis has

to be performed. Assumptions such as rigid bodies, perfect revolute joints,

complete isolation between electrical phenomena and mechanical motions andidealized torque transmission in the gear trains are commonly made [1-5]. By

removing one or several of these assumptions, one can come up with models with

different levels of fidelity. The price to pay is the increased complexity in

the resulted equations and possible numerical difficulties. But sometimes,

the price has to be paid in order to obtain equations that correspond betterwith the important characteristics of the actual robot dynamics. In this

paper, the effect of some idealizations of joint drive systems in the commonly

used model will be investigated.

Although many robot joints are driven by motors through the use of gearswith reasonably high (on the order of hundred) reduction ratio, the commonly

used model does not include any detail of the drive system. Strictly

speaking, the model of the generally used multi-body system is only directlyrelated to a direct drive robot. For this mode] to be applied to a robot with

torque transmission and amplification, certain rules are usually implied. It

is generally believed that the torque at the joints are equal to the product

of the corresponding motor torques and reduction ratios. It is also known tosome that the rotational inertias of the motor rotors when amplified by the

corresponding reduction ratios squared should be included in the link inertiato resist motion. These implications can be found in textbooks on automatic

control, such as [6], but they are rarely spelled out in robot literature and

their validity is not established.

,_.............. ....._ :r;.(_i _-_MED 297

In this paper, a simple model w111 be used to study the effect of gearreduction on system dynamics. Specifically, the necessary modifications tothe equations of motion for the commonly used simply jointed robot model willbe presented. An example will also be used to demonstrate the effects of gearreduction to the equations of motion.

System Model

In the present study, it is assumed that the motor rotor is the onlymassive element in a joint drive system that will contribute to the modifica-

tions of the equations of motion from that of a multi-body direct drive

system. To study this problem, one might be tempted to just study a par-

ticular system by including the rotors in the model and deriving equations forthe system. Simulation can then be performed to hopefully reveal the effects

caused by the inclusion of rotors in the model. This investigation, however,will be only specific for the particular system simulated and will not shed

too much light for a system with very different geometry and mass distribu-

tion. The methodology adopted here is to understand the general effects of

the rotors and thus to enumerate the suitable modifications of the equations

due to them. In place of a specific model, a generic system model should be

used and the resulted equations studied to identify and generalize the effectsof rotors.

The generic model chosen is shown in Fig. 1. Body C is the carrier of

the motor whose rotor together with the connecting shaft and attached gearforms a rigid body R. The driven link D has an attached gear which meshes

with the gear in R. This is an idealized system of two links with simple

drive system. Since it is assumed that the rotor is the only massive element

in the drive system, this model is sufficiently comprehensive for the study.

Instead of letting C be a link jointed to the base, it is allowed a generalmotion relative to the base. This is an intentional choice so that C can beany link of a multi-link system. The system can therefore be considered as a

subsystem of an overall system. It can be seen that the linkage in Fig. Iinvolves a closed-loop topology and formulations for the commonly studied

open-loop multi-link system cannot be applied here. In particular, Newton-

Euler formulation is not very convenient for this system and is especiallydifficult to generalize. Lagrange's or Kane's formulation is more suitable

for the present study because generalized active and inertia forces can be

considered as being contributed from individual elements of the system. The

overall system can be thought of as having n generalized coordinates, ql, -..,qn, and q, the joint angle between C and D, can be one of them.

Formulation of Dynamic Equations

Vector-dyadic formalism will be used in all the following formulations.Here, a vector and a dyadic are defined as abstract entities which are

invariant with respect to unit vector bases [7-9]. They can be expressed in

terms of any unit vector basis or bases but some operations among them can bereduced without being expressed in any basis. Vectors and dyadics as used in

the following are therefore different from column matrices and square matri-

ces, which are just congregates of numbers. However, when the vectors and

the dyadlcs are represented in the same vector basis, their operations can befacilitated by matrix operations.

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The first step in Lagrange's formulation is to derive the kinetic energy

of the system. For the subsystem in Fig. 1, the contribution to the kinetic

energy is

i _ m(NvP) 2 (I)K = _ C+R+D

where N is an inertia reference frame, m and NvP are, respectively, the mass

and the velocity in N of a generic particle P in the system, and _ denotesC+R+D

summatlon over all particles in bodies C, R and D. In the notation for a

velocity, an acceleratlon, an angular velocity or an angular acceleration, a

left superscript is used to denote the reference frame the quantity is referredto. In the sequel, when the left superscript is omitted in any of these nota-

tions, reference frame N is implied. Applying the theorem called one point

moving on a rigid body [9], one can express the velocity of a generic particle P

of R or D by

vP = vC + CvP (2)

where CvP is the veloclty of P in C, and vC is the velocity of _, which is a

point of C that colncldes with P at the instant under consideration.Substitution of equation (2) in equation (I) yields

K = ½ { _ m(vP) 2 . _. mE(CvP) 2 * 2 v_ • CvP]}E R+D

(3)

where E Is a fictitious rlgid body that moves exactly like C but has exactlythe same mass distribution as that of C, R and D altogether at the instant

under conslderatlon.

Wlth the application of a kinematic theorem for two points fixed on a

rlgld body [9], one can express the velocity of a generic particle P of E as

vP = vQ + wc x rQP (4)

where wc is the angular velocity of C in N, Q is an arbitrary reference point

on C and rQP is the position vector from Q to P. The use of equation (4)

brings the first term in the right hand side of equation (3) to the form of

m(vP) 2 wc IEIQ wc vQ rQE*)= mE(vQ)2 + • • + 2mE • (wc x (5)E

where

IEIQ = _ m [(rQP) 2 U - rQP rQP] (6)E

and, mE and E* are the mass and the mass center of E, respectively, while rQE*

is the position vector from Q to E*, and U is a unit dyadic. It should be

299

noticed that E* and IEIQ are not fixed in C• As for the second term in

equation (3), the following equations can be similarly derived by suitablychoosing a reference point• Firstly,

CvR*)2 CwR IR/R * CwRm(CvP) 2 = mR( + • •R

(7)

where

IR/R* = _ m[(rR*P) 2 U - rR*P rR*P]R (8)

C *and, mR and R* are the mass and the mass center of R, while vR and C@R are the

velocity of R* in C and the angular velocity of R in C, respectively, and rR*Pis the position vector from R* to P. Next,

m(CvP)2 = CwD • ID/Q' • CwD (9)D

where

ID/Q' = _ m[(rQ'P) 2 U - rQ'P rQ'P]D

and, rQP and CwD are the position vector from Q', a point on the Joint axis,

to P and the angular velocity of D in C, respectively. Then, making use ofequation (4) for vP, one can write

(10)

m vC • CvP = vQ • mR CvR* + wc • CHR/QR

(11)

where

CHR/Q = _ m r QP x CvPR

and CHR/Q is the angular momentum of R about Q in C• Similarly,

(12)

• C * •T_m vC • CvP = vQ mD vD + wc CHD/QD

(13)

where

CHD/Q = 7_m rQP x CvP

D(14)

Since R* is fixed in C which implies

CvR = 0

(15)

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and

CwR = _ q a

where _ is the gear ratio, it follows that

(16)

and

CHR/Q = p J q a

CwR . IR/R* . CwR = p2 j _2

(17)

(18)

where J is the axial moment of inertia of R. Also since

CwD = q c

equation (9) can be rewritten as

m(CvP)2 = _2 (c. ID/Q' • c)

D

(19)

(20)

Substitution of equations (4)-(18) in equation (3) yields

K = ½ {mE(vQ)2 + _c . IE/Q . wc + _2(p2 j + c • ID/Q' • c)}

+ _c . (rQE* x mEvQ + p J q a + CHD/Q) + mDvQ • CvD* (21)

It can be seen in equation (21) that the kinetic energy of the system involves

Ionly two terms, _ p2j_2 and pJq wc • a that depend exclusively on the motorrotor R. Other contributions to K due to R are lumped in those terms

involving the fictitious body E.

Consider now another system S' consisting of only two links C' and D'

jointed together as shown in Fig. 2. The kinetic energy of S' will be thesame as that in equation (21) less the two terms mentioned above if C, D and E

are replaced by C', D' and E', respectively, and if E' is a fictitious body

having the mass distribution of C' and D' at the instant under consideration.

With perfect rotation of axisymmetrlc rotor R, the inertia dyadic of C and R

together for Q is fixed C. Therefore a real rigid body C' can have the samemass distribution as C and R at all times and have the same motion as C. If

this choice is made, and in addition, D' is chosen to have the mass distribu-

tion and the motion of D, then E' has the same mass distribution as that of E

at all times. The kinetic energies K' and K , respectively, of S' and the

original system S consisting of C, D and R can thus be related by

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(22)

It is worth noticing that the common wisdom of simply adding p2 j to the

"inertia of the driven link" to compensate for the drive system dynamics isonly true when either wc, the angular velocity of the carrier of the drive

system, or wc • a is zero. For many robots there are some drive systems thatdo not satisfy either condition.

The differences between generalized inertia force contributions due to S

and S' can be worked out based on equation (22). Since generalized inertia

force Fr is related to kinetic energy K by

Fr =-(-_t 8_r- Bq_r) r = I, .... , n (23)

it follows that

_ WV

(F) = (F }s' + G r = i, .... , nr s r r

where

_ p j(p_ + ac . a) (qr = q)

pJ{_ B(wc • a) d a(wc • a) B(wc- + 6 [ • a)]}

B_r dt B_r Bqr

(qr _ q)

The partial differentiations in equation (25) are most advantageously per-

formed with C being the reference frame because a is fixed in C. Furthermore,if C is the Ith link in an n-llnk articulated robot as shown in Fig. 3 and Dis the subsequent link, then it can be shown that

a(wc • a) = [zr • a (r _ i)

BQr o (r > i)

B(w C • a) = CBmC • a = {(w Br x Zr) • a

Bq r Bq r 0

(r< i)

(r > i)

(24)

(25)

(26)

(27)

and

302

CddE (Zr " a) : (CwBr x Zr) • a : (Bif_ Br x Zr) • a

With equations (26)-(28) in equation (25), one can rewrite Gr as

+ NaBi • a)- NJ (_ qi+l (r = i + I)

Gr = - _J[(z r • a) qi+l - qi+l (NmBi x Zr) • a] (r _ i)

0 (r > i + I)

(28)

(29)

If the generalized inertia forces of system S' have been worked out separa-

tely, then that of S can be derived using equations (24) and (29).

As to the generalized active forces, consider that R is acted upon

through electromagnetic or viscous damping interaction by C and the net resultof this interaction is a couple of torque Ta. The laws of dynamics dictate

that a couple of torque -Ta is also acted on C by R. The contribution of this

pair of couples to the generalized active forces Fr are simply

NT (r : i + 1)

Fr =0 (r _ i + I)

(3O)

No other interaction forces between the bodies in S contribute to the genera-

lized active forces. For system S', it is easily seen that if a couple of

torque NTc is assumed to act on D' by C', then the contributions of the

interaction forces to the generalized active forces are the same as that in

equation (30).

Generalization

In some situations, the joint drive system of a particular joint ismounted on the outward link rather than the inward link of the Joint, such as

that of the second joint of Unimation PUMA robots. The equations derived forS can still be applied with D being the inward link and C being the outward

link. Consider that C is still the Ith link, and D is the (i-1)th link. The

difference terms Gr in equation (24) become

- _J [N qi + NaBi " a + 41 (zi • a) - qi(Nm Bi x zi) • a] (r : i)

Gr = - _J [(z r • a) qi - qi (NwBi x Zr) • a] (r < i) (31)

o (r> i)

where unit vector a should be in the direction that Is associated in the right

hand sense with the rotation of R when the joint experiences a positive

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rotation. The above generalization is true because all the derivations for

the generic system in Fig. 1 do not depend on whether C or D is the precedinglink, with the exception of equations (16) and (19). When D is the llnk that

precedes C, unit vectors a and c can be properly changed to maintain the

validity of these equations. With this in mind, equation (22) remain validfor the new case. The difference from the original case is that wc in

equation (22) is a function of q for the present case, and it therefore gives

rise to the differences between the expressions in equations (29) and (31).

It is also observed that the above development applies to any axisym-metric body designated R that is carried on C and performs fixed axis rotation

in C. Any gear in a gear train connecting a motor to the link it drives can

be the rotor R and its contribution to the change of the generalized inertiaforces can be identified. It should be noticed that there will be a different

gear ratio and a different unit vector a for each gear.

Furthermore, one can see that the only role D plays in equations (29) or(31) is related to the definition of q which is used in equation (16). If the

generalized coordinates can be properly introduced for the system and the

angular velocity of R in C can be expressed as a function of these generalizedcoordinates, then the role of D can be eliminated. The above results can thusbe extended to the drive system for linear joint. They can also be extended

to complicated gear systems that make up many robot wrist mechanisms such as

that discussed in [10]. For such a system with three degrees of freedom,

C_R = (pl_I + P2_2 + P3_3 ) a (32)

where ql, q2 and q3 are the generalized coordinates associated with the systemand Pl, P2 and P3 are the corresponding ratios for R, should be used instead

of equation (16). With this, equation (22) is replaced by

Ks = KS'+ ½ j(pl_1 + P2_2 + P3_3)2 + j(pl_l + P2_2 + P3_13) _c . a

where S is the system consisting of C and R while S' consists of C' only.(33)

For a complete n degree of freedom motor-driven robot with speed reduc-

tions involved, the above procedure can be applied in the following manner:

1. Model the system as made up of n rigid bodies connected by the

appropriate linear or revolute joints with each of these bodies having themass distribution of the actual link and any rotational elements that itcarries. Formulate the equations of motion for such a model.

2. For each of the rotational elements, figure out its additional contri-

butions to the kinetic energy and in turn the contributions to the generalized

inertia forces as developed above. Add these contributions to the equationsof motion.

3. Use equation (30) to work out the generalized active forces.

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Significance of Reduction Effects

Dynamic equations for PUMA 560 robot has been explicitly derived withparameter values measured or estimated in [11]. Based on the parameter values

listed, the effects of the drive systems can be estimated. For PUMA 560

robots, the 2nd and 3rd joint drive systems are mounted on the 2nd link while

the 4th to 6th drive systems are mounted on the 3rd link. The motors are

mounted in such a way that their axes of rotation are always perpendicular to

the 2nd and the 3rd joint axes. If only the inertia matrix, which is the

congregate of the coefficients of qi, i= I,..,6, and the motor rotor's contri-butions are considered, the coefficients to have additional terms include the

diagonal elements as well as those with indices (1,i), i=2,..., 6, and the off-

diagonal elements with indices (4,5), (4,6) and (5,6) due to the coupling of

the drive systems of the wrist joints. It is understood that the inertia

matrix is a symmetric matrix, and only the elements in the upper triangular

part of the matrix are addressed.

Listed in [11] are inertia contributions of joint drive systems to the

diagonal elements of the inertia matrix. Here, these are assumed to be mainlycontributed by the motor rotors in the form of _2j pursuant to equations

(29) and (30). With this assumption, the constant coefficients of the domi-

nant terms, involving sine and cosine functions of joint angles, of the matrix

elements effected can be compared with those additional contributions propor-

tlonal to _J as computed based on equations (29) and (30). Table 1 shows thelist.

Table 1 Effects of Drive System on Inertia Matrix

Element 1,1 2,2 3,3 4,4 5,5 6,6 1,2 1,3 1,4 1,5 1,6

Dom. Coef. 2.57 6.79 1.16 .20 .18 .19 .69 .13 1.64 1.25 4.

as in [2] E-3 E-3 E-5

Coef. of 1.14 4.71 .83 .2 .18 .19 .04 .015 2.60 2.5 2.5

add. term E-3 E-3 E-3

It can be seen from Table 1 that _2j terms are dominant in the diagonal

elements of the inertia matrix. They are included in the equations in [11]while the additional contributions to the off-diagonal elements that are

proportional to _J are not included. Due to the fact that their contributions

remain constant when the robot posture is changed, the percentage variations of

the diagonal elements is smaller than what it would be if the robot is a direct

drive one. This makes fixed gain control more likely to succeed. Although some

of the latter ones are dominant, they are still very small compared to the

(1,1), or even the (4,4), (5,5) or (6,6) element. As to the elements (4,5),(4,6) and (5,6), since the coupling relationship between the drive systems are

not discussed in [11], the additional contributions cannot be estimated. It is

reasonable to predict that they will be more significant than those for other

off-diagonal elements judging from equation (33).

Contributions due to the motor rotors to those terms second order in ql,

i=1,.., 6, can also be estimated. Again, some of them may be dominant in

the coefficient of a particular term, but their effect to the complete

305

equations may not be significant unless ql, i=1,.., 6, assume significantlyhigh values•

Conclusions

Although it has been known to joint drive system designers that theinertia properties of the motor rotor and other elements connected to it are

important factors in determining the system dynamic response, it has not beenelaborated in so many articles on robot dynamics• The contributions in theform of _2j in the inertia matrix can dominate some of the matrix elements.

Other contributions proportional to _J are less significant, but they may not

be negligible in all cases• With more and very different robots to be developedin the future, it is important to know what need to be included in the dynamicmodel for it to have sufficient fidelity. The methodology set forth in this

paper provides a means to gain the necessary information for a sound judgment.

Acknowledgement

Funding support from NASA GSFC (Grant No. NAGS-1019) for the work

reported is greatly appreciated• The technical officer monitoring the grantis John Celmer.

References

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1984, pp. 371-375.

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on Robotics and Automation, San Francisco, CA, April, 1986, pp. 510-518.

• Q' mrl,,

Q \ ilo 1 IZn

• _-'c___• Qn. 1_---._......._

I71 OnLink Gear Reduction 3Fig. ] lwo System With Z2

zl Q2q D'

I- ° _"---_ ) 7,X;'$?,.,C'

Fig. 3 Multi-Link System

Fig. 2 Two Link Direct Drive System

307